Petr Van ek
Mensur Omerba i
Department of Geodesy and Geomatics Engineering
University of New Brunswick
Abstract
Complications are known to arise when Kalman filter is used in the design of navigation algorithms.
Many improvements have been introduced over the past thirty years. Still, problems seem to occur
when navigation based on a single source data is attempted. Such navigation is then somewhat
dependable upon the knowledge of the vehicle's dynamics model. This knowledge is almost never
available at a level to satisfy the basic reliability requirements.
We seek a solution through an approach different from that of Kalman filtering in so far that no
physical modelling is called for. We use a purely mathematical approach based on the laws of
mechanics (e.g., Hamilton's principle), applied in a statistical space. The result is a new navigation
algorithm (NNA) that relies solely upon the measurements of vehicle's positions (and possibly
velocities), and their error statistics. The force field in which non-Newtonian forces are at work, is
induced simply by confidence regions of the position fixes. It is assumed that instantaneous position
fixes and their corresponding error statistics can be obtained by one or more navigation services, such
as GPS or Loran C . One promissing application of the NNA that will be investigated in the near
future is the tracking of evasive targets based on position and/or velocity measurements. The results
of the new algorithm are presented for a couple of real-life trajectories and a comparison with the best
Kalman filter navigation is shown.
1
Introduction
Since the early 1960's, modern navigation has been making use of the hybrid (integrated)
navigation systems, where various electronic sensing devices (sensors) are used side by side, to
collect the information necessary to find the "continuous" position of the navigated vehicle.
These systems are installed on-board of vehicles, such as ships, aircraft, or missiles. Sensors that
are being used in such systems are the Inertial Navigation Systems (INS), radio-navigation aids
(LORAN, GPS, etc.), Doppler Velocity Sensors (DVS), laser-ranging devices, barometric
altitude-meters, etc.
The potential benefits from multi-sensor system implementation lead to a development of
mathematical algorithms for merging the data from various sources (sensors). The automation
of continuous positioning, and the decrease of direct human involvement in the positioning
process became important considerations. These developments lead to a new general concept of
navigation: the computational solutions were sought in algorithms that make use of filtering. At
the beginning of the sixties, a Hungarian (electrical) engineer Rudolf Kalman introduced a new
kind of filtering, where the observations relating to the motion of the vehicle, as well as a
dynamic model derived from some physical laws, were required. Consequently, the success of
1
this filter to predict the correct position depends to a large extent on the validity of both kinds of
information.
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Almost forty years after it was first introduced, the most widely applied navigation algorithms
are still based on Kalman filter. Here, we give several examples that highlight the desirability of
replacing Kalman filter in the navigation systems used in marine vessels, aircraft and missiles,
by a more appropriate algorithm. We will discuss these examples as they also serve the purpose
of putting the new navigation algorithm in the proper context:
1) As a part of aerodynamics in military applications synthesis, it has been known for some
time that the development of highly agile aircraft requires an understanding of the unsteady
flow associated with these aircraft. The a priori aerodynamic force and moment input used
for the design of these aircraft are studied by dynamic experiments in wind tunnels. Such
experiments include both oscillatory and rotary techniques capable of testing at extreme
attitudes and/or high angular rate motions. These motions represent a highly variable and
unpredictable dynamic environment.
2) In the context of geodesy and geomatics, a vehicle or missile, equipped with the INS/GPS
navigation device(s) [Coffee & Maganty, 1996] requires a specific algorithm (filter) to
perform the integration of an Inertial Navigation System (INS) with the Global Positioning
System (GPS). A need for such integration is well documented by, e.g., Kleusberg [1988]: it
should be performed so as to allow an at-all-times reliable navigation solution. For instance,
the simple navigation filters built-in the GPS receivers are dependent on the vehicle
dynamics model when GPS alone is used for navigation purposesthey require model
set/reset input from the pilot/navigator as soon as a significant state change has occurred.
Another source of problems lays in the unavoidable loss of the GPS signal, due to
maneuvering (masking). A model-less algorithm, capable of self-following and guidance of
the vehicle under all-conditions maneuvering is then preferable.
3) In marine navigation, the problems appear when trying to model tidal and other ocean
currents, or the atmospheric variations. Here the difficulty in applying the Kalman filter lays
in the fact that it requires an a-priori system model to be given; in marine navigation any
such model can never be realistic. An algorithm that will do the self-following and
guidance of the vessel under all-conditions maneuvering is thus required.
4) When employed in a combat situation, a fighter jet's navigation system based on Kalman
filter does not perform very well if input data are based on position/velocity measurements.
One of the reasons for this is that at such time the aircraft navigation system suffers the
"memory loss", i.e., the model dependent filter needs a new model to cope with the fast
developing situation. Such a model is, of course, not available and this results in a
navigation failure. A model-less filter, capable of self-following and guidance under allconditions maneuvering is needed.
2
5) In (military) industry, the need to re-design algorithms based on the Kalman filter, to run
efficiently on every new platform (application) can be very costly. An adaptable, platformindependent algorithm could be highly desirable.
6) Tracking a target that employs evasive maneuvering could be very unreliable when the
tracking algorithm is based on a Kalman filter. Particularly so, if the input for the algorithm
consists only of position and/or velocity measurements.
2
Mathematical background of the new navigation algorithm
It is known from dynamics that the motion of a system in which all forces (except those derived
from constraints) can be derived from a single generalized potential function that depends on
coordinates, velocities and time. This concept is implied by, e.g., the Hamilton's principle
[Boccaletti & Pucacco, 1996]. We define the generalized potential function in such a way that it can
be constructed solely from the position fixes and their confidence regions, and call it the position
potential. We assume that instantaneous position fixes and the corresponding error statistics for
the navigated vehicle are obtained by one or more navigation services, such as GPS or Loran C.
Adopting the above concept, in its kinematic form, the acceleration is given by:
a i = ri =
∂U
,
∂ ri
(1)
where U is the potential of the force field we want the navigated vehicle to obey. The navigated
vehicle is then represented simply by a mass particle. The notion of "position potential" enables
us to define that force field which drives the free particle along its trajectory.
Conceptually, the solution uses the force field in which non-Newtonian forces (i.e., forces that are
not inversely proportionate to squared distances) are induced by the confidence regions – called
here also force sources – that pull the (undetermined) trajectory of the particle (Fig.1) to pass
through them. In fact, the solution is built on a time-varying position potential field U, defined as
a summation of n individual position potential fields defined by the quadratic forms of the
confidence regions. Thus
U (r, t; α , G ) = Ge −α t
n
i =1
(r − ri0 ) T C i−1 (r − ri0 ) ⋅ eα ti , t ≥ tn
(2)
with
r=
x
y
and
ri0 =
x0
,
y0
(3)
being the position vectors (here two-dimensional, but can be extended into three dimensions) of
the particle and of the i-th position fix, respectively, both considered at time ti. Here G is a
positive scale factor, analogous to the gravitational constant in the Newtonian mechanics, and α
is the temporal attenuation coefficient. Both numbers, G and α , can be either selected
beforehand, or evaluated during the navigation process from the fit of the predicted trajectory to
new position fixes, following one of many possible schemes. Different applications may require
different approaches to the selection and updating of these two parameters to ensure the optimum
performance of the algorithm. So far only a few simple approaches have been tried on the
trajectories we had at our disposal [Xu, 1996]. Further investigation is clearly indicated here.
3
The equation of motion is then obtained from eqn.(2) by differentiation with respect to ri:
∂U
= ri = − e −αt (A ri − B)
∂ ri
(4)
where
n
A = 2G ⋅
i =1
n
B = 2G ⋅
i =1
eα ( ti −tn ) Ci−1
e
.
α ( ti −tn )
(5)
−1 0
i i
C r
For t ≥ 0 , this happens to be the second order Bessel differential equation of order 0 [Xu, 1996],
the solution (ri) of which is given by Bessel functions of first kind. For more information on the
mathematical background and previous work on the new algorithm, the interested reader is
referred to Xu (1996).
3
Summary of test results
Tests made with the NNA consisted of examining its performance on both synthetic and real two
dimensional trajectories, while considering various schemes for determining the parameters G
and α . The best performing scheme turned out to be a pre-selection of a specific value of α ,
suitable for the kind of tested motion accompanied by continuous updating of the G value based
on the "past experience" of the algorithm. The results were then compared with the best Kalman
filter (KF) navigation.
As shown in Fig.3, the NNA can indeed be regarded as a filtering technique. Clearly, the most
accurate results are obtained when the algorithm is used in its "smoothing mode", while the
"estimation mode" gives worse results, and the "prediction mode" performs the worst. The NNA
represents also a capable blunder detector - it picked up two artificially implanted blunders, at the
0.001 confidence level.
A comparison of the performance of the NNA with that of the KF was done over two real ship
trajectories. The NNA performed, on average, three to four times better than the KF, i.e., with an
improvement of 60 to 80%. The results are shown in Fig.2 and summed up in Table 1. We note
that the speed of processing by the NNA is slower than that by the KF, by a factor 5 to 10. This
drawback of the NNA could be alleviated by using parallel processing or some other advanced
computing methodology which were not at our disposal during the testing.
Table 1: comparative results of KF and NNA tests – speed vs. accuracy
0.02 sec
0.11, 0.23 sec
4
8 m
2 m
3 m
1 m
To summarize, the NNA has the following advantages. It does not require any model of vehicle's
dynamic environment, and, consequently, it does not depend on the platform it is mounted on; it
suffers from the phenomenon of "overshooting" much less than the KF-based algorithm does; as
Kalman filter, NNA can also be used for detecting blunders in position fixes; and can be easily
modified to accept also velocity information. Its disadvantage is in its slower processing speed.
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Discussion
At its present form, the New Navigation Algorithm (NNA) is a 2-D self-learning algorithm (G,
and possibly α , are being continuously updated), flexible and responsive to a changing dynamic
environment, with minimal overshooting (Fig.2). It represents a unique, platform-independent
navigation tool, capable of multi-sensor information integration. Tested with both simulated and
real navigation data, it proved to be a valid estimator, predictor, smoother and blunder detector
(Fig.3). Compared with the Kalman filter, the new algorithm requires the uncertainties of
observations to be known only relatively, as cofactor matrices. Further, the NNA allows a direct
input and real time integration of position and/or velocity measurements (taken at the vehicle, i.e.,
it is independent of the ground control or navigator's input) to speed up the response of the
algorithm to changing dynamics.
The new filter "learns" (i.e. updates the model parameters regularly) all the time, hence it can
keep pace with the kinematic change of the vehicle and thus achieve better results. It appears that
the use of this algorithm would be more appropriate than that of the Kalman filter for navigating
vehicles that are expected to maneuver under their own power. The inclusion of velocity
observations in this algorithm results in a more accurate navigation than that using either
observed positions alone, or velocities alone (dead reckoning). The performance of the new
algorithm appears to be sturdy enough to detect possible blunders in position fixes. Compared
with the Kalman filter for the vehicle under maneuvers, the new algorithm achieves better results,
but its processing speed is slower.
5
Conclusions
The introduction of the new navigation algorithm would have positive consequences for
navigation. These consequences are:
-
increased safety, as the new algorithm would result in a navigation that is stable, safe and
reliable under all conditions, and
-
increased economy, as the algorithm is platform independent.
So far, the algorithm is formulated only for two-dimensional navigation, but an expansion into
three dimensions can be achieved relatively easily so that its application in aviation can be
realistically contemplated. Further improvement of the new algorithm can be expected from a
selection of an optimal "position potential function" (to replace the generic "position potential
5
function" used in our investigations so far), which requires more experimental testing. Because
of algorithm's high flexibility, it is possible to envision an improvement to its self-learning
capabilities.
References
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Antoulas, A.C. (1991). Mathematical System Theory: The Influence of R.E. Kalman.
Springer-Verlag, Berlin, Germany.
Boccaletti, D., G. Pucacco (1996). Theory of orbits: Integrable systems and nonpertubative methods. Springer-Verlag, Rome, Italy.
Gutman, P.O. and Velger, M. (1990), “Tracking Targets Using Adaptive Kalman
Filtering.” IEEE Transactions on Aerospace and Electronic Systems, Vol. 26, No. 5, pp.
691-698.
Inzinga, T. and Van ek, P. (1985). “A two-dimensional Navigation Algorithm Using a
Probabilistic Force field.” Presented at Third International Symposium on Inertial
Technology for Surveying and Geodesy, Banff, Canada, 1985.
Kleusberg, A. (1988). "Integrated INS/GPS navigation". Paper presented at the
International Workshop in Stuttgart and Altensteig, Germany. Published in the
workshop's proceedings: Linkwitz, K., U. Hangleiter, (Eds.) (1988): High Precision
Navigation, Stuttgart, Germany.
Van ek, P. and Krakiwsky, E.J. (1986). Geodesy: the concepts. Second Edition,
Elsevier Science Publishers B.V. 1992.
Xu, B. (1996). “A New Navigation Filter.” Technical Report No. 182, The Department of
Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton, NB
Canada.
Figures
Figure 1 Figure 2 Figure 3 -
History of trajectory evolution.
Comparison between the NNA and the KF test results.
Time history of actual position errors for one of the real ship trajectories.
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Figure 1: History of trajectory evolution: confidence regions (error ellipses) of position fixes tend
to attract the undetermined trajectory to pass through them. The impact of the force sources
diminishes with time (as the trajectory evolves). The position fix with smaller confidence region
(the more accurate fix) attracts more than a fix with larger confidence region. The closer the
position fix to particle position (at the time of the fix) the weaker the attracting force.
Figure 2: Comparison between the new navigation algorithm and the Kalman filter test results
(approximately scaled)
8
Figure 3: Time history of actual position errors for one of the real ship trajectories
9